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Problem 11

# Denote by $$\mathcal{K}$$ the class of functions $[0, \infty) \rightarrow[0, \infty)$ which are zero at zero, strictly increasing, and continuous, by $$\mathcal{K}_{\infty}$$ the set of unbounded $$\mathcal{K}$$ functions, and by $$\mathcal{K} \mathcal{L}$$ the class of functions $[0, \infty)^{2} \rightarrow[0, \infty)$$which are of class$$\mathcal{K}$ on the first argument and decrease to zero on the second argument. A continuous-time time-invariant system $$\dot{x}=f(x, u)$$ with state space $$X=\mathbb{R}^{n}$$ and control-value space $$\mathcal{U}=\mathbb{R}^{m}$$ is said to be input-to-state stable (ISS) if there exist a function $$\beta \in \mathcal{K} \mathcal{L}$$ and a function $$\gamma \in \mathcal{K}_{\infty}$$ so that the following property holds: For each $$T>0$$, if $$\omega$$ is a control on the interval $$[0, T]$$ and $$\xi$$ is a solution of $$\dot{\xi}=f(\xi, \omega)$$ on the interval $$[0, T]$$, then $$\|\xi(T)\| \leq \max \left\\{\beta(\|\xi(0)\|, T), \gamma\left(\|\omega\|_{\infty}\right)\right\\}$$

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To summarize, a continuous-time, time-invariant system $$\dot{x}=f(x, u)$$ with state space $$X=\mathbb{R}^{n}$$ and control-value space $$\mathcal{U}=\mathbb{R}^{m}$$ is Input-to-State Stable (ISS) if there exist functions $$\beta \in \mathcal{K} \mathcal{L}$$ and $$\gamma \in \mathcal{K}_{\infty}$$ such that the following property holds for any given $$T>0$$, control function $$\omega$$ on the interval $$[0, T]$$, and solution $$\xi$$ of the differential equation $$\dot{\xi}=f(\xi, \omega)$$ on interval $$[0, T]$$: $$\|\xi(T)\| \leq \max \left\{\beta(\|\xi(0)\|, T), \gamma\left(\|\omega\|_{\infty}\right)\right\}$$
See the step by step solution

## Step 1: Understanding the Class of Functions $$\mathcal{K}$$

The class of functions, $$\mathcal{K}$$, consists of functions of the form $$[0, \infty) \rightarrow [0, \infty)$$, which means that the domain and range of these functions are from 0 to infinity. Additionally, the functions in this class are strictly increasing, continuous, and zero at zero, i.e., they have the properties: 1. $$f(0) = 0$$ 2. Continuous on the domain $$[0, \infty)$$ 3. Strictly increasing

## Step 2: Understanding the Class of Functions $$\mathcal{K}_\infty$$

The class of functions $$\mathcal{K}_\infty$$ consists of unbounded functions in $$\mathcal{K}$$. This means that these functions have all the properties of $$\mathcal{K}$$ functions (as described in Step 1) and are also unbounded, which means there is no upper limit on their range.

## Step 3: Understanding the Class of Functions $$\mathcal{K}\mathcal{L}$$

The class of functions $$\mathcal{K}\mathcal{L}$$ consists of functions of the form $$[0, \infty)^2 \rightarrow [0, \infty)$$. These functions have two arguments and their output ranges from 0 to infinity. Additionally, these functions have the following properties: 1. They are of class $$\mathcal{K}$$ on the first argument, which means they have the properties of $$\mathcal{K}$$ functions when only considering the first argument. 2. They decrease to zero on the second argument, which means that, as the second argument increases, the value of the function approaches zero.

## Step 4: Understanding Input-to-State Stability (ISS)

A continuous-time, time-invariant system, represented by the differential equation $$\dot{x}=f(x, u)$$ with state space $$X=\mathbb{R}^{n}$$ (i.e., it has n states) and control-value space $$\mathcal{U}=\mathbb{R}^{m}$$ (i.e., it has m controls), is said to be Input-to-State Stable (ISS) if the following conditions exist: 1. A function $$\beta \in \mathcal{K} \mathcal{L}$$. 2. A function $$\gamma \in \mathcal{K}_{\infty}$$. These functions must satisfy the given property.

## Step 5: The Property of Functions $$\beta$$ and $$\gamma$$

For the system to be ISS, the following property must hold for any given $$T>0$$, control function $$\omega$$ on the interval $$[0, T]$$, and solution $$\xi$$ of the differential equation $$\dot{\xi}=f(\xi, \omega)$$ on interval $$[0, T]$$: $$\|\xi(T)\| \leq \max \left\\{\beta(\|\xi(0)\|, T), \gamma\left(\|\omega\|_{\infty}\right)\right\\}$$ This inequality describes the bounded behavior of the solution $$\xi(T)$$ in terms of the initial state $$\xi(0)$$, the time $$T$$, and the control function $$\omega$$.

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